Theorem tgbtwntriv2 25799

Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
tgbtwntriv2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 787	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥))
2		tkgeom.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		tkgeom.d	. . . . . 6 ⊢ − = (dist‘𝐺)
4		tkgeom.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
5		tkgeom.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG)
7		tgbtwntriv2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	7	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃)
9		simplr 785	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃)
10		simpr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵))
11	2, 3, 4, 6, 8, 9, 8, 10	axtgcgrid 25775	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥)
12	11	adantrl 707	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥)
13	12	oveq2d 6921	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
14	1, 13	eleqtrrd 2909	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15		tgbtwntriv2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
16	2, 3, 4, 5, 15, 7, 7, 7	axtgsegcon 25776	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)))
17	14, 16	r19.29a 3288	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5013

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-trkgc 25760  df-trkgcb 25762  df-trkg 25765

This theorem is referenced by:  tgbtwncom  25800  tgbtwntriv1  25803  tgcolg  25866  legid  25899  hlid  25921  lnhl  25927  tglinerflx2  25946  mirreu3  25966  mirconn  25990  symquadlem  26001  outpasch  26064  hlpasch  26065